Boy or girl paradox

The initial formulation of the question dates back to at least 1959, when Martin Gardner featured it in his October 1959 "Mathematical Games column" in Scientific American.

He titled it The Two Children Problem, and phrased the paradox as follows: Gardner initially gave the answers ⁠1/2⁠ and ⁠1/3⁠, respectively, but later acknowledged that the second question was ambiguous.

The ambiguity, depending on the exact wording and possible assumptions, was confirmed by Maya Bar-Hillel and Ruma Falk,[3] and Raymond S.

[4] Other variants of this question, with varying degrees of ambiguity, have been popularized by Ask Marilyn in Parade Magazine,[5] John Tierney of The New York Times,[6] and Leonard Mlodinow in The Drunkard's Walk.

[2] This answer is intuitive if the question leads the reader to believe that there are two equally likely possibilities for the sex of the second child (i.e., boy and girl),[2] and that the probability of these outcomes is absolute, not conditional.

[8] First, it is assumed that the space of all possible events can be easily enumerated, providing an extensional definition of outcomes: {BB, BG, GB, GG}.

[9] This notation indicates that there are four possible combinations of children, labeling boys B and girls G, and using the first letter to represent the older child.

[9] This implies the following model, a Bernoulli process with p = ⁠1/2⁠: Under the aforementioned assumptions, in this problem, a random family is selected.

In response to reader criticism of the question posed in 1959, Gardner said that no answer is possible without information that was not provided.

Specifically, that two different procedures for determining that "at least one is a boy" could lead to the exact same wording of the problem.

But they lead to different correct answers: Grinstead and Snell argue that the question is ambiguous in much the same way Gardner did.

[10] They leave it to the reader to decide whether the procedure, that yields 1/3 as the answer, is reasonable for the problem as stated above.

This ambiguity leaves multiple possibilities that are not equivalent and leaves the necessity to make assumptions about how the information was obtained, as Bar-Hillel and Falk argue, where different assumptions can lead to different outcomes (because the problem statement was not well enough defined to allow a single straightforward interpretation and answer).

It must be further assumed that Mr. Smith would always report this fact if it were true, and either remain silent or say he has at least one daughter, for the correct answer to be ⁠1/3⁠ as Gardner apparently originally intended.

Three of the four equally probable events for a two-child family in the sample space above meet the condition, as in this table: Thus, if it is assumed that both children were considered while looking for a boy, the answer to question 2 is ⁠1/3⁠.

However, if the family was first selected and then a random, true statement was made about the sex of one child in that family, whether or not both were considered, the correct way to calculate the conditional probability is not to count all of the cases that include a child with that sex.

"[11] Another way to analyse the ambiguity (for question 2) is by making explicit the generative process (all draws are independent).

The paradox arises because the second assumption is somewhat artificial, and when describing the problem in an actual setting things get a bit sticky.

However, this one is equivalent to "sampling" the distribution (i.e. removing one child from the urn, ascertaining that it is a boy, then replacing).

If so, as combination BB has twice the probability of either BG or GB of having resulted in the boy walking companion (and combination GG has zero probability, ruling it out), the union of events BG and GB becomes equiprobable with event BB, and so the chance that the other child is also a boy is ⁠1/2⁠.

In 1991, Marilyn vos Savant responded to a reader who asked her to answer a variant of the Boy or Girl paradox that included beagles.

With regard to her survey they say it "at least validates vos Savant's correct assertion that the "chances" posed in the original question, though similar-sounding, are different, and that the first probability is certainly nearer to 1 in 3 than to 1 in 2."

The authors conclude that, although the assumptions of the question run counter to observations, the paradox still has pedagogical value, since it "illustrates one of the more intriguing applications of conditional probability.

Again, the answer depends on how this information was presented – what kind of selection process produced this knowledge.

To understand why this is, imagine Marilyn vos Savant's poll of readers had asked which day of the week boys in the family were born.

This variant of the boy and girl problem is discussed on many internet blogs and is the subject of a paper by Ruma Falk.

However, this does not exhaust the boy or girl paradox for it is not necessarily the ambiguity that explains how the intuitive probability is derived.

A survey such as vos Savant's suggests that the majority of people adopt an understanding of Gardner's problem that if they were consistent would lead them to the ⁠1/3⁠ probability answer but overwhelmingly people intuitively arrive at the ⁠1/2⁠ probability answer.

Ambiguity notwithstanding, this makes the problem of interest to psychological researchers who seek to understand how humans estimate probability.

The authors argued that the reason people respond differently to each question (along with other similar problems, such as the Monty Hall Problem and the Bertrand's box paradox) is because of the use of naive heuristics that fail to properly define the number of possible outcomes.